ME 3560 Fluid Mechanics

ME3560 – Fluid Mechanics

Chapter VIII. Viscous Flow in Pipes

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ME 3560 Fluid Mechanics




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8.1 General Characteristics of Pipe Flow•Most of the conduits used to transport fluid from one location toanother are round in cross section.

•Conduits such as typical water pipes, hydraulic hoses, and otherconduits are designed to withstand a considerable pressure differenceacross their walls without undue distortion of their shape.

•Typical conduits of noncircular cross section include heating and airconditioning ducts that are often of rectangular cross section.

•Normally the pressure difference between the inside and outside ofthese ducts is relatively small.



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•In the following study, it is assumed that the pipe is completely filledwith the fluid being transported.

•Thus, a concrete pipe through which rainwater flows withoutcompletely filling the pipe is not considered here

•Such flows are called open-channel flow.

•The difference between open-channel flow and the pipe flow of thischapter is in the fundamental mechanism that drives the flow.

•For open-channel flow, gravity alone is the driving force—the waterflows down a hill.

•For pipe flow, gravity may be important but the main driving force islikely to be a pressure gradient along the pipe. If the pipe is not full, it isnot possible to maintain this pressure difference.



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8.1.1 Laminar or Turbulent Flow•Osborne Reynolds (1842–1912), a British scientist and mathematician,was the first to distinguish the difference between turbulent and laminarflow

•Reynolds injected dye into a pipe in which water flowed due to gravity.•If water runs through a pipe of diameter D with an average velocity V,the following characteristics are observed by injecting neutrally buoyantdye as shown.



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•These three characteristics, denoted as laminar, transitional,and turbulent flow.



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•For pipe flow the most important dimensionless parameter is theReynolds number, Re—the ratio of the inertia to viscous effects in theflow.

DV

Re

•Re ranges for which laminar, transitional, or turbulent pipe flows areobtained cannot be precisely given. The actual transition from laminar toturbulent flow may take place at various Reynolds numbers, dependingon how much the flow is disturbed by vibrations of the pipe, roughnessof the entrance region, and the like.•For general engineering purposes (i.e., without undue precautions toeliminate such disturbances), the following values are appropriate:

Laminar Flow: Re < 2100Turbulent Flow: Re > 4000



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8.1.2 Entrance Region and Fully Developed Flow•The region of flow near where the fluid enters a pipe is termedthe entrance region.•The fluid typically enters the pipe with a nearly uniform velocityprofile at section (1).



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•A boundary layer in which viscous effects are important is producedalong the pipe wall such that the initial velocity profile changes withdistance along the pipe, x, until the fluid reaches the end of the entrancelength, section (2), beyond which the velocity profile does not varywith x.•The boundary layer has grown in thickness to completely fill the pipe.Viscous effects are of considerable importance within the boundarylayer.



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•The shape of the velocity profile in the pipe depends on whether theflow is laminar or turbulent, as does the length of the entrance region, le.As with many other properties of pipe flow, the dimensionless entrancelength, le/D, correlates quite well with the Reynolds number. Typicalentrance lengths are given by

6/1(Re)4.4

Re06.0

DlDl

e

e Laminar Flow

Turbulent Flow•For low Re flows the le can be short (ℓe = 0.6D if Re = 10)•For large Re flows le can be equal to many pipe diameters beforereaching the end of the entrance region. (ℓe = 120D for Re = 2000).•For many practical engineering problems, 104 < Re < 105 so that

20D < le < 30D.



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8.2 Fully Developed Laminar Flow•By solving the NS–Equations it was found that for a SS, Laminar flowin a pipe (Poiseuille flow)

•Now, consider a differential element of fluid in a fully developed flowdriven by pressure gradient under SS–conditions in a pipe (Poiseuilleflow)

LpR

xpRu

Rruru

44

1)(

22

max

2

2

max



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Clpru

lpr

drdu

lpr

lrrlrpprp

Fx

sin4

)sin(2

)sin(2

0sin)2()(

0

2

221

21

•At r = R, u = 0:

lpRC

sin

4

2

lpRu

Rr

lpRru

sin

41sin

4)(

2

max

22Then:

2rp

2)( rpp

2

2

max 1)(Rruru



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sin8

21

4

0 2

2

max

lpRQ

rdrRruudAQ

R

)sin(2

)sin(2

max

lpR

lpr

wall

•Consider now the Energy Equation

2121

2

222

1

211

22

zzpph

hzg

Vphzg

Vp

L

Ls

sin21 lzz

sinlphL



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sinsin

lpllphL

)sin(2

lpR

wall

•Then:

•Since, as determined previously:

wallLl

Rh

2



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8.3 Fully Developed Turbulent Flow

•Flows are classified as laminaror turbulent.•As a general rule for pipeflow, if Re < 2100 the flow islaminar, whereas if Re > 4000the flow is turbulent.

8.3.1 Transition from Laminar to Turbulent Flow

•A trace of the axial componentof velocity measured at a givenlocation in the flow, u = u(t), isshown•Its irregular, random nature isthe distinguishing feature ofturbulent flow.



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•The fundamental differencebetween laminar and turbulentflow lies in the chaotic, randombehavior of the various fluidparameters.•Such variations occur in thethree components of velocity,the pressure, the shear stress,the temperature, and any othervariable that has a fielddescription.•Turbulent flow is characterizedby random, three-dimensionalvorticity.

8.3.2 Turbulent Shear Stress

•Turbulent flows can be described interms of their mean values on whichare superimposed the fluctuations.

'

),,,(1 0

0

uuu

dttzyxuT

uTt

t

0'_u

•Notice that:



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u

dtuT

uu

Tt

t

2/12______

20

0

)'(1)'(

T

•Define the Turbulence Intensity (T ):

•Notice that in turbulent flows:yu

•Instead the shear stress is defined in terms of the laminar shear stressand the turbulent shear stress: turbulentlaminar

are called Reynolds Stresses and they are positive

_____

''vuyu

_____

''vu• In a very narrow regionnear the wall (the viscoussublayer), the laminarshear stress is dominant.



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•Away from the wall (inthe outer layer) theturbulent portion of theshear stress is dominant.• The transition betweenthese two regions occursin the overlap layer.

•Due to the difficulties to determine the Reynolds stresses, a number ofmodels have been proposed to calculate τturb•Boussinesq Model

yu

turb

• η is known as the Eddy viscosity coefficient

yulm

2• Prandtl’s Mixing Length Model:

•lm is the mixing length, thus: 22

yulmturb



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•Fully developed turbulent flow in a pipe can be broken into threeregions which are characterized by their distances from the wall:

8.3.3 Turbulent Velocity Profile

• Viscous sublayer very near thepipe wall.• Overlap region.• Outer turbulent layerthroughout the center portion ofthe flow.

•Within the viscous sublayer thefluid viscosity is an importantparameter; the density isunimportant.•In the outer layer the opposite istrue.



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• Viscous sublayer:

*

*

yuuu

5/0 * yuu* is called friction velocity.

y = R – r.

2/1*

wu

.

(Law of the wall)

• Overlap Region:

5ln5.2*

*

yu

uu

13/5 * yu• Outer Layer:

RruuV

yR

uuV

c

c

/11ln5.2

ln5.2

*

*

Vc Centerline velocity.



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• Another commonly used relation for the velocity in turbulent flow is thepower – law velocity profile: n

c Rr

Vu /1

1

• n is function of the Re.•The one-seventh power-law velocityprofile (n = 7) is often used as areasonable approximation for manypractical flows.



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8.4 Dimensional Analysis of Pipe Flow• Applying the energy equation to a pipe system requires thedetermination of the head loss (hL) in the system.•A typical pipe system usually consists of various lengths of straight pipeinterspersed with various types of components (valves, elbows, etc.).•hL for a pipe system consists of the head loss due to viscous effects inthe straight pipes, termed the major loss, and the head loss in the variouspipe components, termed the minor loss.

minormajor LLL hhh8.4.1 Major Losses•The pressure drop and head loss in a pipe are dependent on the wallshear stress, τw, between the fluid and pipe surface.•A fundamental difference between laminar and turbulent flow is that theshear stress for turbulent flow is a function of the density of the fluid.• For laminar flow, the shear stress is independent of the density, leavingthe viscosity, μ, as the only important fluid property.



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• For steady, incompressible turbulent flow in a horizontal round pipe ofdiameter D:

),,,,,( lDVfp • Applying dimensional analysis:

k = 7, r = 3

),,(

21 1

2 DDlDV

V

p

• By realizing that Δp is proportional to the lengthof the pipe:

),(

21 2 D

DVDl

V

p

• If a friction factor is defined as: )(Re,D

f

DlVfp

2

2



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Lhzg

Vpzg

Vp 2

22

22

1

21

11

22

• From the energy equation for a horizontal pipe with constant diameterand under steady state:

gV

DlfpphL 2

221

major

gV

DlfhL 2

2

major Darcy–Weisbach equation:

• In general, for non–horizontal pipes:

2)()(

2

121221V

Dlfzzhzzpp L



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• For laminar flow (horizontal pipes):lpD

lpRV

328

22

232 2

2

VDlf

DVlp

DDVf

Re6464

• But we found that:

• For turbulent flow, the friction factor is obtained from Moody diagramof from empirical correlations.

fD

f DRe51.2

7.3/log0.21 • Colebrook’s Equation:

D

Df Re

9.67.3

/log8.11 1.1• Modified Colebrook’sEquation:

DlVfp

2

2

• Thus, combining the two previous equations:



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• Moody Diagram:



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8.4.2 Minor Losses•Pipe systems consist of a number of fittings: valves, bends, tees, etc. •The losses added to the system for such fittings are termed minor losses.

gVKh LL 2

2

minor • KL minor loss coefficient

• The previous relation is also presented as:

gV

Dl

fh eqL 2

2

minor Dl

fK eqL

• D and f are based on the pipe containing the component.

•Values for KL are tabulated.

•Thus, the pressure losses in a system can be determined by using oneof Colebrook’s equations or the Moody Diagram.



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8.4.3 Non – Circular Ducts•For laminar flow in non–circular ducts the friction factor is written as

•C depends on the particular shape of the duct•Reh = ρVDh/μ• Dh is the hydraulic diameter•Dh = 4A/P is four times the ratio of the cross-sectional flow area divided by the wetted perimeter, P.

h

CfRe

[For Round Pipes: Dh = 4A/P = 4(πD2/4)/(πD) = D]

• hL = f(ℓ/Dh)V2/2g.•Relative roughness, ε/Dh.



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• Calculations for fully developed turbulent flow in ducts of noncircular cross section are usually carried out by using the Moody chart data for round pipes with the diameter replaced by the hydraulic diameter and the Reynolds number based on the hydraulic diameter.

h

CfRe

• hL = f(ℓ/Dh)V2/2g.

• Thus, for laminar flow:



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8.5 Examples• In the analysis of single pipe systems three different type of problems occur:For all three cases ε, ρ, and μ are known:1. Type I: V, L, D are known. Find Δp and/or hL.

2. Type II: D, Δp are known. Find V and/or Q.

3. Type III: V, Δp are known. Find D.

Types II and III require iteration.



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gVK

gV

DLfh

hzg

Vpgm

Wzg

Vp

LTotalL

TotalLs

22

2222

2

22

22

1

21

11

• Thus, to analyze the flow in single pipes the following relations need to be applied:



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ME 3560 Fluid Mechanics